Transformations of instances of collection - problem with "origin"
I don't think your coordinate space is at fault per se, I think you need to mess with your operating sets.  By description alone it sounds like you have your active Transform Pivot to be "Individual Origins", "Median Point" sounds like what you want.



You can mess with your working translation space by clicking the "global" directly to the left and choosing something else.

Something else you might want to do is parent all your pieces of your object to something, either the root mesh, an armature, or an Empty.  This will allow you to conveniently move and animate the whole set by moving the parent.  

Lastly: if you mirror by duplicating and scaling by -1, you will also need to flip the normals depending on usage.  It is oftentimes better to use the Mirror modifier instead, it is more convenient and stable.